How to calculate the numerical integration that contains singular points？

T(x,y,afa) is a generated integrand, and the codes are as following.When I calculate M=arrayfun(@(D) integral2(@(x,y) T(x, y, D), 0,pi/2,-pi/6,pi/6,'reltol', 1e-6), afa) with varying afa=0:0.005:pi/6, the curve of output is not smooth and seems like noise. This is because the integrand has singular points. How to solve this problem? Many thanks!

function U=T(x,y,afa)
d1=1.34e-9;
d2=1.34e-9;
mu=5.5;
vh=1;
HBAR=1.05457266e-34;
ME=9.1093897e-31;
ELEC=1.60217733e-19;
Kh=2.95e10;
kc=sqrt(2.*ME.*ELEC./HBAR.^2);
k=kc.*sqrt(mu);
kh=sqrt(k.^2-(Kh-k.*sin(x).*cos(y)).^2-k.^2.*sin(x).^2.*sin(y).^2);
khg=sqrt(k.^2-(2.*Kh.*sin(afa./2).*sin(afa./2)-k.*sin(x).*cos(y)).^2-(2.*Kh.*sin(afa./2).*cos(afa./2)+k.*sin(x).*sin(y)).^2);
khpl=sqrt(k.^2-(Kh-k.*sin(x).*cos(y)).^2-k.^2.*sin(x).^2.*sin(y).^2+kc.^2.*vh);
khplpl=sqrt(k.^2-(Kh-k.*sin(x).*cos(y)).^2-k.^2.*sin(x).^2.*sin(y).^2+2.*kc.^2.*vh);
khgplpl=sqrt(k.^2-(2.*Kh.*sin(afa./2).*sin(afa./2)-k.*sin(x).*cos(y)).^2-(2.*Kh.*sin(afa./2).*cos(afa./2)+k.*sin(x).*sin(y)).^2+2.*kc.^2.*vh);
A2=exp(i.*khpl.*d1)./(exp(i.*(kh+khgplpl-khg).*d1)+exp(i.*khplpl.*d1));
U=abs(A2).^2;
end

Answer 1

[Raynier Suresh]

The “quadgk” function can handle singularity if the singularity is present at the boundary. In case if your singularity is not at the boundary you can split the integration domain to place the singularity at the boundary. Refer to the below links for more information,

Quadgk: https://www.mathworks.com/help/matlab/ref/quadgk.html#References

Split integration domain: https://www.mathworks.com/help/matlab/math/singularity-on-interior-of-integration-domain.html